Instant Mental Calculation of Fifth Roots

Fifth Root Calculation

Learn how to mentally calculate fifth roots and amaze your friends!

Following a few simple steps you will be able to instantly calculate the fifth root of the spectator's number.

Ask the spectator to choose any whole number less than 100 and, using a calculator, to find its cube by multiplying the number by itself, then multiplying the answer by the original number.

Alternatively, if the spectator knows how to use scientific calculator, they can enter the original number, then press the x^y key followed by 3 and =.

The spectator then calls out the answer, and you instantly reveal the original number (i.e., the cube root).

Note that this method will NOT work with non-whole numbers, or with numbers greater than 99.

Step 1: Learn the Fifth Powers of 0 to 9

To master the system you must learn by heart the fifth powers of numbers 0 to 9, which are shown in the table below. You also need to consider the last digit of each fifth power.

Number

5th Power

Last Digit

0

0

0

1

1

1

2

32

2

3

243

3

4

1024

4

5

3125

5

6

7776

6

7

16807

7

8

32768

8

9

59049

9

Note how the last digit of every fifth power ends with the original number.

Step 2: Determine the Fifth Root

Ignore the last five digits of the number called out by the spectator and choose the memorised fifth power which is just lower (or equal) to the remaining number. The corresponding fifth root is the first digit of your answer.

Now consider the last digit of the number called out by the spectator. This is the last digit of your answer (see table above).

Example 1

Spectator calls 1073741824

Ignore last five digits = 10737

Lower (or equal) fifth power = 7776

Fifth root of 7776 is 6 = first digit of answer

Last digit of spectator's call is 4 = last digit of answer

Therefore the fifth root of 1073741824 is 64

Example 2

Spectator calls 130691232

Ignore last five digits = 1306

Lower (or equal) fifth power = 1024

Fifth root of 1024 is 4 = first digit of answer

Last digit of spectator's call is 2 = last digit of answer

Therefore the fifth root of 130691232 is 42

Further Examples

CalledFifthPower

IgnoreLastFive

Lower(or =)Power

FirstDigit(F)

LastDigitof Call

LastDigit(S)

FifthRoot= FS

248832

2

1

1

2

2

12

17210368

172

32

2

8

8

28

69343957

693

243

3

7

7

37

282475249

2824

1024

4

9

9

49

418195493

4181

3125

5

3

3

53

777600000

7776

7776

6

0

0

60

1804229351

18042

16807

7

1

1

71

4704270176

47042

32768

8

6

6

86

7339040224

73390

59049

9

4

4

94

Practice, Practice, Practice!

Before you try this out on your friends, you should practice until you can calculate the fifth root instantly and without error.

If you can't do this, then practice some more!

To practice and assess your ability, you can use the Fifth Root Tester shown below.